[R] polynomials REML and ML in nlme

[Douglas Bates]

Spencer Graves wrote:
>  Am I correct that changing the parameterization should NOT change the 
> estimates of the variance components, because you are minimizing 
> essentially the same objective function over the same subspace? The only 
> thing that changes is the logdet(X'WX) term mentioned by I White? 
> Moreover, letting X = QR, and using the fact that det(AB) = 
> det(A)*det(B) if they are both square, we get det(R'Q'WQR) = 
> det(Q'WQ)*det(R)^2. Thus, the change in parameterization affects only R, 
> not Q, which means that it can't affect det(Q'WQ).
> 
> Is this accurate?
> 
> As a simple sanity check, I ran a very simple model with the same fixed 
> effects in different parameterizations, and got the same estimates for 
> the variance components under REEL but different 
> "log-restricted-likelihood" (see below).
> 
> Comments?

You are correct - it is only the R part that changes.  If you were to 
require an orthonormal parameterization of the fixed effects then the 
REML criterion would be invariant with respect to linear parameter 
transformations.  That's essentially the same as using the ML deviance 
instead of the REML deviance.

Before his untimely death Greg Reinsel was working on likelihood ratio 
tests using the REML criterion instead of the likelihood.  You need to 
adjust the criterion for the submodel relative to the full model but the 
pieces for doing that are sitting around in the lmer object.  I plan to 
include that in a future version of the lme4 package.

Meanwhile your idea looks quite interesting.  It may be possible to 
define the REML criterion in such a way that it produces the same 
estimates as now but does not change with linear transformations of the 
fixed effects.